RO-TST 2014 Problem 10

Let $ABC$ be an isosceles triangle, $AB = AC$ , and let $M$ and $N$ be points on the sides $BC$ and $CA$ , respectively, such that $\angle BAM=\angle CNM$ . The lines $AB$ and $MN$ meet at $P$ . Show that the internal angle bisectors of the angles $BAM$ and $BPM$ meet at a point on the line $BC$ .